Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwo point function for critical points of a random plane wave
220
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.
rate research
Read More
We consider the $n$-component $|varphi|^4$ lattice spin model ($n ge 1$) and the weakly self-avoiding walk ($n=0$) on $mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+alpha)}$ with $alpha in (0,2)$. The upper critical dimension is $d_c=2alpha$. For $epsilon >0$, and $alpha = frac 12 (d+epsilon)$, the dimension $d=d_c-epsilon$ is below the upper critical dimension. For small $epsilon$, weak coupling, and all integers $n ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-alpha)}$. This sticking of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.
Consider the long-range models on $mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-alpha}$ for some $alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $alpha e2$ that, assuming a bound on the derivative of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{alphawedge2-d}$ above the upper-critical dimension $d_cequiv(alphawedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $alpha=2$ decays as $|x|^{2-d}/log|x|$ whenever $dge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.
We consider long-range self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)asymp|x|^{-d-alpha}$ with $alpha>0$. The upper-critical dimension $d_{mathrm{c}}$ is $2(alphawedge2)$ for self-avoiding walk and the Ising model, and $3(alphawedge2)$ for percolation. Let $alpha e2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{alphawedge2-d}$, where the constant $Cin(0,infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $alpha<2$ and $alpha>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.
We introduce and study the following model for random resonances: we take a collection of point interactions $Upsilon_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schrodinger Hamiltonians $H_Upsilon = -Delta + ``sum mathfrak{m}(alpha) delta (x - Upsilon_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $Sigma (H_Upsilon)$ in the complex plane $mathbb{C}$. We show that the counting function for the set of random resonances $Sigma (H_Upsilon)$ in $mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $Upsilon$, and obtain an explicit formula for the limiting distribution as $m to infty$ of the leading parameter of the asymptotic chain of `most narrow resonances generated by a sequence of uniform binomial processes $Upsilon^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.
We study a pair of infinite dimensional dynamical systems naturally associated with the study of minimizing/maximizing functions for the Strichartz inequalities for the Schrodinger equation. One system is of gradient type and the other one is a Hamiltonian system. For both systems, the corresponding sets of critical points, their stability, and the relation between the two are investigated. By a combination of numerical and analytical methods we argue that the Gaussian is a maximizer in a class of Strichartz inequalities for dimensions one, two and three. The argument reduces to verification of an apparently new combinatorial inequality involving binomial coefficients.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
